Question

[{secA-tanA}/secA+tanA}]^1/2 = [cosA/{1+sinA}]

Answer 1

Formula Applied :

$\sf \bullet\ \; sec\theta=\dfrac{1}{cos\theta}\\\\\bullet\ \; \sf tan\theta=\dfrac{sin\theta}{cos\theta}\\\\\bullet\ \; \sf sin^2\theta+cos^2\theta=1$

Solution :

LHS

$\sf \to \sqrt{\dfrac{secA-tanA}{secA+tanA}}\\\\\to \sf \sqrt{\dfrac{\frac{1}{cosA}-\frac{sinA}{cosA}}{\frac{1}{cosA}+\frac{sinA}{cosA}}}\\\\\to \sf \sqrt{\dfrac{\frac{1-sinA}{cosA}}{\frac{1+sinA}{cosA}}}\\\\\to \sf \sqrt{\dfrac{1-sinA}{1+sinA}}$

Rationalize the numerator

$\to \sf \sqrt{\dfrac{1-sinA}{1+sinA}\times \dfrac{1+sinA}{1+sinA}}\\\\\to \sf \sqrt{\dfrac{1-sin^2A}{(1+sinA)^2}}\\\\\to \sf \sqrt{\dfrac{cos^2A}{(1+sin^2A)}}\\\\\to \sf \dfrac{cosA}{1+sinA}$,

RHS

Answer 2

Answer:

The Fundamental Duties' are defined as the moral obligations of all citizens to help promote a spirit of patriotism and to uphold the unity of India. These duties set out in Part IV–A of the Constitution, concern individuals and the nation. Like the Directive Principles, they are not enforceable by courts unless otherwise made enforceable by parliamentary law.